Find Amplitude, Period, and Phase Shift y=1+3sin(2x-pi/2)

$y = 1 + 3 \sin (2 x - \frac{π}{2})$

Step 1

Rewrite the expression as $3 \sin (2 x - \frac{π}{2}) + 1$ .

$3 \sin (2 x - \frac{π}{2}) + 1$

Step 2

Use the form $a \sin (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = 3$

$b = 2$

$c = \frac{π}{2}$

$d = 1$

Step 3

Find the amplitude $| a |$ .

Amplitude: $3$

Step 4

Find the period using the formula $\frac{2 π}{| b |}$ .

Find the period of $3 \sin (2 x - \frac{π}{2})$ .

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

Find the period of $1$ .

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

The period of addition/subtraction of trig functions is the maximum of the individual periods.

$π$

Step 5

Find the phase shift using the formula $\frac{c}{b}$ .

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{\frac{π}{2}}{2}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{2} \cdot \frac{1}{2}$

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

Multiply $\frac{π}{2}$ by $\frac{1}{2}$ .

Phase Shift: $\frac{π}{2 \cdot 2}$

Multiply $2$ by $2$ .

Phase Shift: $\frac{π}{4}$

Step 6

List the properties of the trigonometric function.

Amplitude: $3$

Period: $π$

Phase Shift: $\frac{π}{4}$ ( $\frac{π}{4}$ to the right)

Vertical Shift: $1$

Step 7

Find Amplitude, Period, and Phase Shift y=1+3sin(2x-pi/2)

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